A weather radar uses a radio antenna that emits short bursts of radio waves. These radio waves travel through the atmosphere until they hit something, such as rain or snow. The wave is then reflected back to the antenna, which measures the time it took for the wave to return.
Between each burst emitted from the antenna, there is a “listening” period in which it receives these reflected waves. Generally, a stronger reflected wave indicates a more reflective surface, such as larger raindrops or hail.
A Doppler radar uses the Doppler Effect to measure how fast the reflective surface is moving toward or away from the antenna. The Doppler Effect is the change in frequency of a wave for an observer moving relative to its source. This is similar to what happens when you hear an ambulance coming towards you. It sounds higher pitched; then right in front of you it sounds normal; then as it passes you, it sounds lower. The wavelength of the sound being emitted is always the same, but as the ambulance moves toward you each part of the wave takes slightly less time to reach you, and therefore you perceive the sound as higher.
Similarly, if rain is moving toward the antenna, the reflected wave will be slightly higher in frequency than the original radio wave. If rain is moving away from the antenna, the reflected wave will be slightly lower in frequency than the original radio wave. How much higher or lower the frequency of the reflected wave is will tell you how fast the rain is moving.
Keep in mind that radar images (like infrared images) are not pictures of the actual data the instrument collects—the data a weather radar collects are a series of measurements of the reflected waves. That data can tell us how far away precipitation is, how heavy it is, and how fast it is moving, which we can then represent on a map.